To move forward I will offer to summarise the simple points I was making, which I'm willing to admit maybe totally wrong.
#1 Empathy is the ability to put yourself in someone's shoes, and sympathy is a feeling of goodwill towards a person.
#2 You can only have sympathy, if you already have empathy for a person. Not vice versa.
On statement #1 we do not disagree. In statement #2, I have said that there is a pre-condition for sympathy, i.e. you must feel what the other person is feeling on some level before you can have sympathy or antipathy toward them. This seems like common sense to me, but you have taken issue with it. As it turns out the website you posted has a technical definition for empathy which totally separates it from sympathy. I don't understand how this is possible, so I presented my view as summarized above.
Actually, I can offer a counter explanation. It maybe possible to have sympathy if you have been conditioned by society to feel for a person in pain despite not really having empathy for them. My counter-counter explanation is that on some level you are still empathizing with their pain in an abstract sense, whether or not you can relate to the situation which brings about the pain.